In this paper, we determine and count the left, right and two-sided cells within \(W_2\)  for all \(\mathbf {a}(2)\) -finite
Coxeter systems. Our results center around the notion of stubs. To define them, recall that every pair of
noncommuting generators \(I=\lbrace i,j\rbrace \subseteq S\)  of a Coxeter group \(W\)  gives rise to to four partially defined functions
on \(W\) , namely, the left upper, left lower, right upper and right lower star operations (see
sec:StarOperations). These operations generalize the star operations introduced in [1]}. We define a
left stub in \(W\)  to be an element that admits no right lower star operation with respect to any pair of
noncommuting generators; similarly, we define a right stub to be an element admitting no left lower star
operation. We equip the set of left stubs in \(W_2\)  with two equivalence relations, one called slide
equivalence and the other called simple slide equivalence (def:equivalences), and we will show that stubs and
these two relations “control everything” about the cells of \(W_2\)  in \(\mathbf {a}(2)\) -finite Coxeter systems.
